Trialities of orthosymplectic $\mathcal{W}$-algebras

Abstract

Trialities of $\mathcal{W}$-algebras are isomorphisms between the affine cosets of three different $\mathcal{W}$-(super)algebras, and were first conjectured in the physics literature by Gaiotto and Rapčák. In this paper we prove trialities among eight families of $\mathcal{W}$-(super)algebras of types $B$, $C$, and $D$. The key idea is to identify the affine cosets of these algebras with one-parameter quotients of the universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra which was recently constructed by Kanade and the second author. Our result is a vast generalization of both Feigin-Frenkel duality in types $B$, $C$, and $D$, and the coset realization of principal $\mathcal{W}$-algebras of type $D$ due to Arakawa and us. It also provides a new coset realization of principal $\mathcal{W}$-algebras of types $B$ and $C$. As an application, we prove the rationality of the affine vertex superalgebra $L_k({\mathfrak{osp}}_{1|2n})$, the minimal $\mathcal{W}$-algebra ${\mathcal{W}}_{k-1/2}({\mathfrak{sp}}_{2n+2},f_{\text{min}})$, and the coset $\text{Com}(L_k({\mathfrak{sp}}_{2m}),L_k({\mathfrak{sp}}_{2n}))$, for all integers $k,n,m\ge 1$ with $m<n$. We also prove the rationality of some families of principal $\mathcal{W}$-superalgebras of ${\mathfrak{osp}}_{1|2n}$ and ${\mathfrak{osp}}_{2|2n}$, and subregular $\mathcal{W}$-algebras of ${\mathfrak{so}}_{2n+1}$

Some corrections and expository improvements, references added, final version to appear in Advances in Mathematics

References in corpus (4)

• Braided tensor categories and extensions of vertex operator algebras
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• Higher rank FZZ-dualities
• Symplectic level-rank duality via tensor categories

Cited by in corpus (12)

• Quantum toroidal algebras and solvable structures in gauge/string theory
• Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras
• Cosets of free field algebras via arc spaces
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• $N=1$ super Virasoro tensor categories
• Cosets from equivariant W-algebras
• Orthosymplectic Feigin-Semikhatov duality
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• Feigin-Semikhatov conjecture and related topics
• Duality via convolution of W-algebras